### Increased computational accuracy in multi-compartmental cable models (Lindsay et al. 2005)

Accession:129149
Compartmental models of dendrites are the most widely used tool for investigating their electrical behaviour. Traditional models assign a single potential to a compartment. This potential is associated with the membrane potential at the centre of the segment represented by the compartment. All input to that segment, independent of its location on the segment, is assumed to act at the centre of the segment with the potential of the compartment. By contrast, the compartmental model introduced in this article assigns a potential to each end of a segment, and takes into account the location of input to a segment on the model solution by partitioning the effect of this input between the axial currents at the proximal and distal boundaries of segments. For a given neuron, the new and traditional approaches to compartmental modelling use the same number of locations at which the membrane potential is to be determined, and lead to ordinary differential equations that are structurally identical. However, the solution achieved by the new approach gives an order of magnitude better accuracy and precision than that achieved by the latter in the presence of point process input.
Reference:
1 . Lindsay AE, Lindsay KA, Rosenberg JR (2005) Increased computational accuracy in multi-compartmental cable models by a novel approach for precise point process localization. J Comput Neurosci 19:21-38 [PubMed]
Model Information (Click on a link to find other models with that property)
 Model Type: Neuron or other electrically excitable cell; Brain Region(s)/Organism: Cell Type(s): Channel(s): I Na,t; I K; Gap Junctions: Receptor(s): Gene(s): Transmitter(s): Simulation Environment: NEURON; C or C++ program; Model Concept(s): Methods; Implementer(s):
Search NeuronDB for information about:  I Na,t; I K;
 / LindsayEtAl2005 readme.txt 03-192.pdf AnalyseResults.c BitsAndPieces.c CellData.dat CompareSpikeTrain.c Ed04.tex ExactSolution.dat GammaCode Gen.tex Gen1.tex Gen2.tex Gen3.tex Gen4.tex Gen5.tex Gen6.tex GenCom.c GenCom1.c GenCom2.c GenComExactSoln.c GenerateInput.c GenerateInputText.c GenRan.ran GetNodeNumbers.c Info100.dat Info20.dat Info200.dat Info30.dat Info300.dat Info40.dat Info400.dat Info50.dat Info500.dat Info60.dat Info70.dat Info80.dat Info90.dat InputCurrents.dat InputDendrite.dat JaySpikeTrain.c JayTest1.dat JayTest100.dat KenSpikeTrain.c KenTest1.dat * KenTest10.dat KenTest100.dat * KenTest10p.dat KenTest1p.dat * KenTest2.dat KenTest2p.dat KenTest3.dat KenTest3p.dat KenTest4.dat KenTest4p.dat KenTest5.dat KenTest5p.dat KenTest6.dat KenTest6p.dat KenTest7.dat KenTest7p.dat KenTest8.dat KenTest8p.dat KenTest9.dat KenTest9p.dat LU.c Mean50.dat Mean500.dat mosinit.hoc NC.pdf NC.tex NC1.tex NC2.tex NC3.tex NC4.tex NC5.tex NC6.tex NCFig2.eps * NCFig3.eps * NCFig4.eps * NCFig5a.eps * NCFig5b.eps * NCFig6.eps * NCPics.tex NeuronDriver.hoc NewComExactSoln.c NewComp.pdf NewComp.ps NewComp.tex NewComp.toc NewComp1.tex NewComp2.tex NewComp3.tex NewComp4.tex NewComp5.tex NewComp6.tex NewCompFig1.eps NewCompFig2.eps * NewCompFig3.eps * NewCompFig4.eps * NewCompFig5a.eps * NewCompFig5b.eps * NewCompFig6.eps * NewCompPics.tex NewComSpikeTrain.c NewRes.dat NewRes60.dat NewRes70.dat NewRes80.dat NewSynRes40.dat NewTestCell.d3 NResults.res OldComExactSoln.c out.res principles_01.tex rand Ratio.dat RelErr.dat ReviewOfSpines.pdf SpikeTimes.dat TestCell.d3 TestCell1.d3 TestCell2.d3 TestCell3.d3 TestCell4.d3 testcellnew2.hoc TestCGS.c TestGen1.c TestSim.hoc TestSim020.hoc TestSim030.hoc TestSim040.hoc TestSim050.hoc TestSim060.hoc TestSim070.hoc TestSim080.hoc TestSim090.hoc TestSim1.hoc TestSim100.hoc TestSim200.hoc TestSim300.hoc TestSim400.hoc TestSim500 TestSim500.hoc
\section{Construction of the model differential equations}
Section \ref{part} showed how the various components of point and
distributed input can be partitioned between the proximal and
distal boundaries of a segment. Once the total axial current
$I_\mathrm{PD}+I_\mathrm{P}$ at the proximal boundary of a segment
and $I_\mathrm{PD}+I_\mathrm{D}$ at the distal boundary of a
segment are determined, the family of ordinary differential
equations modelling the branched dendrite is constructed by
enforcing conservation of current at all segment boundaries.

Each dendritic terminal at which the potential is unknown
contributes one differential equation with form determined by the
properties of the terminal. For example, if the terminal is
sealed, the differential equation expresses the condition
$I_\mathrm{PD}+I_\mathrm{D}=0$. At a dendritic branch point, the
single differential equation is formed by equating the sum of the
proximal current in the child segments to the distal current in
the parent segment. A point soma behaves like a branch point with
the current crossing the somal membrane playing the role of the
distal current in the parent segment. Finally, at all other
segment boundaries, the differential equation is constructed by
equating the distal current of one segment to the proximal current
of its neighbour.

Suppose that there are $m$ nodes at which the potential is
unknown, then the compartmental model of the neuron will be
written for the potentials
$$\label{cmde1} V(t)=\big[\,V_1(t),V_2(t),\cdots,V_m(t)\,]^\mathrm{T}\,.$$
where $V_k(t)$ is the potential at the $k^{th}$ node. The system
of differential equations satisfied by $V(t)$ has general form
$$\label{gmde2} C\,\frac{dV}{dt}+G_\mathrm{SYN}(t)\,V+G_\mathrm{IVDC}(\bs{\theta}(t))\,V-AV+I(t)=0$$
where $C$, $G_\mathrm{SYN}(t)$, $G_\mathrm{IVDC}(\bs{\theta}(t))$
and $A$ are $m\times m$ matrices such that their $(j,k)^{th}$
entry is non-zero whenever the $j^{th}$ and $k^{th}$ nodes lie at
opposite ends of a segment, \emph{i.e.}, they are neighbouring
nodes. In equation (\ref{gmde2}), $A$ is a constant matrix of
axial conductances and $C$ is a constant matrix of capacitances.
The function $G_\mathrm{SYN}(t)$ is a matrix of time-dependent
conductances associated with synaptic input to the dendrite, the
function $G_\mathrm{IVDC}(\bs{\theta}(t))$ is a matrix of
time-dependent conductances associated with intrinsic
voltage-dependent transmembrane current to the dendrite, and
$I(t)$ is a column vector of voltage-independent currents.
Equation (\ref{gmde2}) is integrated over the interval $[t,t+h]$
to get
$$\label{gmde3} \begin{array}{l} \ds C\big[\,V(t+h)-V(t)\big]+\int_t^{t+h} G_\mathrm{SYN}(t)V(t)\,dt+\int_t^{t+h} G_\mathrm{IVDC}(\bs{\theta}(t))V(t)\,dt\$10pt] \qquad\qquad\ds-\;A\int_t^{t+h} V(t)\,dt+\int_t^{t+h} I(t)\,dt=0 \,. \end{array}$$ The trapezoidal rule is used to estimate each integral in equation (\ref{gmde3}) with the exception of the integral of intrinsic voltage-dependent current which is estimated by the midpoint rule. The result of this calculation is $$\label{gmde4} \begin{array}{l} \ds C \big[V(t+h)-V(t)\big]+\frac{h}{2} \Big[G_\mathrm{SYN}(t+h)V(t+h)+G_\mathrm{SYN}(t)V(t)\Big]\\[10pt] \ds\qquad+\;h G_\mathrm{IVDC}\big(\bs{\theta}(t+h/2)\big) V(t+h/2) -\frac{h}{2} \Big[A V(t+h)+AV(t)\Big]\\[10pt] \ds\qquad\qquad\ds+\;\frac{h}{2}\Big[I(t+h)+I(t)\Big] +O(h^3)=0\,. \end{array}$$ By noting that 2V(t+h/2)=V(t+h)+V(t)+O(h^2), equation (\ref{gmde4}) may be rearranged to give $$\label{mde6} \begin{array}{l} \ds \Big[2C-hA+h\,G_\mathrm{SYN}(t+h) +h\,G_\mathrm{IVDC}\big(\bs{\theta}(t+h/2)\big)\,\Big]\,V(t+h) = \\[10pt] \quad\ds \Big[\,2C+h\,A-h\,G_\mathrm{SYN}(t) -h\,DG_\mathrm{IVDC}\big(\bs{\theta}(t+h/2)\big)\,\Big]\,V(t) -h\Big[I(t+h)+I(t)\Big]+O(h^3). \end{array}$$ The computation of G_\mathrm{IVDC}\big(\bs{\theta}(t+h/2)\big) depends on how intrinsic voltage-dependent current is specified. For example, for a membrane following Hodgkin-Huxley kinetics, G_\mathrm{IVDC}\big(\bs{\theta}(t+h/2)\big) is specified in terms of the solutions of a set of auxiliary equations. In this case, it is well known that G_\mathrm{IVDC}\big(\bs{\theta}(t+h/2)\big) can be computed to adequate accuracy from V(t) and the differential equations satisfied by the auxiliary variables (\emph{e.g.}, see Lindsay \emph{et al.}, \cite{Lindsay01a}). The coefficient matrices in equation (\ref{mde6}) are therefore determined by V(t) and known prior to the determination of the potential V(t+h). \subsection{Some additional comments} All compartmental models of a dendrite begin with a subdivision of its sections into contiguous segments. The segments, in turn, define the compartments of the mathematical model. Both the new and traditional compartmental models are based on the \emph{same} morphological segments. In a traditional compartmental model, the distribution of membrane potential throughout a dendrite is described by the membrane potentials at the centres of dendritic segments. By contrast, in the new compartmental model the membrane potential throughout a dendrite is described by the potential at segment endpoints. The number of nodes at which potentials are to be determined, and consequently the numerical complexity of the problem, are identical in both types of compartmental model. Furthermore, both models involve nearest neighbour interactions, and so the structure of the differential equations describing either model is identical. Consequently benefits such as the existence of a sparse matrix factorisation of the matrix on the left hand side of equation (\ref{mde6}) are enjoyed by both types of model. Finally, it should be noted that the development of the new compartmental model highlights structural differences between the treatment of point input in this model and their treatment in a numerical procedure used to solve the partial differential equations of the continuum model. In the compartmental model, conservation of current is applied at each synapse to arrive at an equation connecting potentials at neighbouring nodes. In a numerical procedure (\emph{e.g.}, finite elements or finite differences), the potential at synapses is estimated on the basis of the assumed representation of the potential between nodes. Consequently, numerical procedures often conserve current in an averaged sense, but not necessarily point-wise at a synapse. It is unclear to what extent such a treatment of synaptic input influences the accuracy of numerical schemes. \section{The model neuron} The comparison of the accuracy of the traditional and new compartmental models is based on the construction of a branched neuron for which the continuum model has a closed form expression for the membrane potential in response to exogenous input. This solution then stands as a reference against which the performance of the traditional and new compartmental models can be assessed. The most effective way to construct a branched model neuron with a closed form solution for the membrane potential is to choose the radii and lengths of its sections such that the Rall conditions for an equivalent cylinder are satisfied (Rall, \cite{Rall64}). These conditions require that the sum of the three-halves power of the diameters of the child limbs is equal to the three-halves power of the diameter of the parent limb at any branch point, and that the total electrotonic length from a branch point to dendritic tip is independent of path. In particular, the electrotonic distance from soma-to-tip is independent of path. The model neuron used in our simulation exercises, illustrated in Figure \ref{TestNeuron}, satisfies these conditions. When the Rall conditions are satisfied, the effect at the soma of any configuration of input on the branched model of the neuron is identical to the effect at the soma of the unbranched equivalent cylinder with biophysical properties and configuration of input determined uniquely from those of the original branched neuron (Lindsay \emph{et al.}, \cite{Lindsay03}). \begin{figure}[!h] \[ \begin{array}{c} \includegraphics[ ]{NewCompFig4.eps} \end{array}\qquad \begin{array}{ccc} \hline \mbox{Section} & \mbox{Length }\mu\mbox{m} & \mbox{Diameter }\mu\mbox{m}\\[2pt] \hline (a) & 166.809245 & 7.089751 \\ (b) & 379.828386 & 9.189790 \\ (c) & 383.337494 & 4.160168 \\ (d) & 410.137845 & 4.762203 \\ (e) & 631.448520 & 6.345604 \\ (f) & 571.445800 & 5.200210 \\ (g) & 531.582750 & 2.000000 \\ (h) & 651.053246 & 3.000000 \\ (i) & 501.181023 & 4.000000 \\ (j) & 396.218388 & 2.500000 \\ \hline \end{array}$
\centering
\parbox{5.5in}{\caption{\label{TestNeuron} A branched neuron
satisfying the Rall conditions. The diameters and lengths of the
dendritic sections are given in the right hand panel of the
figure. At each branch point, the ratio of the length of a section
to the square root of its radius is fixed for all children of the
branch point.}}
\end{figure}

%\begin{figure}[!h]
%$%\begin{array}{c} %\begin{mfpic}[1][1]{0}{220}{-20}{220} %\pen{2pt} %\dotsize=1pt %\dotspace=3pt %\lines{(-5,100),(5,110),(15,100),(5,90),(-5 ,100)} %% Upper dendrite %% Root branch %\dotted\lines{(5,115),(15,170),(20,170)} %\lines{(20.0,160),(36.7,160)} %\tlabel[tc](28.4,150){\textsf{(a)}} %% Level 1 %\lines{(50.0,190),(88.3,190)} %\tlabel[bc](75,200){\textsf{(c)}} %\lines{(50.0,130),(91.0,130)} %\tlabel[tc](75,120){\textsf{(d)}} %\dotted\lines{(36.7,160),(45,200),(55,200)} %\dotted\lines{(36.7,160),(45,120),(55,120)} %% Level 2 %\lines{(100.0,210),(153.2,210)} %\lines{(100.0,190),(153.2,190)} %\lines{(100.0,170),(153.2,170)} %\tlabel[cl](160,210){\textsf{(g)}} %\tlabel[cl](160,190){\textsf{(g)}} %\tlabel[cl](160,170){\textsf{(g)}} %\dotted\lines{(88.3,190),(95,220),(105,220)} %\dotted\lines{(88.3,190),(95,160),(105,160)} %\lines{(100.0,140),(165.1,140)} %\lines{(100.0,120),(165.1,120)} %\dotted\lines{(91.0,130),(95,150),(105,150)} %\dotted\lines{(91.0,130),(95,110),(105,110)} %\tlabel[cl](175,140){\textsf{(h)}} %\tlabel[cl](175,120){\textsf{(h)}} %% %% Lower dendrite %% Root branch %\lines{(20.0,40),(58.0,40)} %\dotted\lines{(5,85),(15,30),(25,30)} %\tlabel[bc](39,50){\textsf{(b)}} %% Level 1 %\lines{(70.0,70),(133.1,70)} %\lines{(70.0,10),(127.1,10)} %\dotted\lines{(58,40),(66.5,80),(76.5,80)} %\dotted\lines{(58,40),(66.5,0),(76.5,0)} %\tlabel[bc](105,80){\textsf{(e)}} %\tlabel[tc](105,0){\textsf{(f)}} %% Level 2 %\lines{(145,80),(195.1,80)} %\lines{(145,60),(195.1,60)} %\dotted\lines{(133.1,70),(140,90),(150,90)} %\dotted\lines{(133.1,70),(140,50),(150,50)} %\tlabel[cl](205,80){\textsf{(i)}} %\tlabel[cl](205,60){\textsf{(i)}} %\lines{(140,30),(179.6,30)} %\lines{(140,10),(179.6,10)} %\lines{(140,-10),(179.6,-10)} %\dotted\lines{(127.1,10),(134,40),(144,40)} %\dotted\lines{(127.1,10),(134,-20),(144,-20)} %\tlabel[cl](190,30){\textsf{(j)}} %\tlabel[cl](190,10){\textsf{(j)}} %\tlabel[cl](190,-10){\textsf{(j)}} %\end{mfpic} %\end{array}\qquad %\begin{array}{ccc} %\hline %\mbox{Section} & \mbox{Length }\mu\mbox{m} & \mbox{Diameter }\mu\mbox{m}\\[2pt] %\hline % (a) & 166.809245 & 7.089751 \\ % (b) & 379.828386 & 9.189790 \\ % (c) & 383.337494 & 4.160168 \\ % (d) & 410.137845 & 4.762203 \\ % (e) & 631.448520 & 6.345604 \\ % (f) & 571.445800 & 5.200210 \\ % (g) & 531.582750 & 2.000000 \\ % (h) & 651.053246 & 3.000000 \\ % (i) & 501.181023 & 4.000000 \\ % (j) & 396.218388 & 2.500000 \\ %\hline %\end{array} %$
%\centering
%\parbox{5.5in}{\caption{\label{TestNeuron} A branched neuron
%satisfying the Rall conditions. The diameters and lengths of the
%dendritic sections are given in the right hand panel of the
%figure. At each branch point, the ratio of the length of a section
%to the square root of its radius is fixed for all children of the
%branch point.}}
%\end{figure}

To guarantee that any apparent errors between the closed form
solution and the numerical solution from either compartmental
model are not due to the lack of precision with which the branched
dendrite is represented as an equivalent cylinder, a high degree
of accuracy is used in the specification of dendritic radii and
section lengths in the model neuron. The model neuron illustrated
in Figure \ref{TestNeuron} is assigned a specific membrane
conductance of $0.091\,$mS/cm$^2$ ($g_\mathrm{M}$) and specific
membrane capacitance of $1.0\,\mu$F/cm$^2$ ($c_\mathrm{M}$), and
axoplasm of conductance $14.286\,$mS/cm ($g_\mathrm{A}$). With
these biophysical properties, the equivalent cylinder has length
one electrotonic unit. The soma of the test dendrite is assumed to
have membrane area $A_\mathrm{S}$, specific conductance
$g_\mathrm{S}=g_\mathrm{M}$ and specific capacitance
$c_\mathrm{S}=c_\mathrm{M}$.

\subsection{Analytical solution}
It may be shown that $V(t)$, the deviation of the somal
transmembrane potential from its resting value as a result of a
distribution $\mathcal{I}(x,t)$ of current on a uniform
cylindrical dendrite of radius $a$ and length $l$ attached to a
soma is
$$\label{es4} V(t)=e^{-t/\tau}\,\Big[\,\phi_0(t)+\sum_\beta\;\phi_\beta(t) e^{-\beta^2 t/L^2\tau}\,\cos\beta\,\Big]\,, \qquad L=l\,\sqrt{\ds\frac{2 g_\mathrm{M}}{a g_\mathrm{A}}}$$
where $\tau$ is the time constant of the somal and dendritic
membranes and $g_\mathrm{M}$ and $g_\mathrm{A}$ have their usual
meanings. The summation is taken over all the solutions $\beta$ of
the transcendental equation $\tan\beta+\gamma\beta=0$ where
$\gamma$ (constant) is the ratio of the total membrane area of the
soma to the total membrane area of the dendrite. The functions
$\phi_0(t)$ and $\phi_\beta(t)$ are solutions of the differential
equations
$$\label{es9} \begin{array}{rcl} \ds\frac{d\phi_0}{dt} & = & -\ds\frac{e^{t/\tau}} {C_\mathrm{D}+C_\mathrm{S}}\,\Big[\, \mathcal{I}_\mathrm{S}(t)+\int_0^l\,\mathcal{I}(x,t)\,dx\,\Big]\,,\\[10pt] \ds\frac{d\phi_\beta}{dt} & = & -\ds\frac{2e^{(1+\beta^2/L^2)t/\tau}} {C_\mathrm{D} +C_\mathrm{S}\cos^2\beta}\,\Big[\, \int_0^1\,\mathcal{I}(x,t)\cos\beta\big(1-x/l\big)\,dx +\cos\beta\,\mathcal{I}_\mathrm{S}(t)\,\Big] \end{array}$$
with initial conditions $\phi_0(0)=\phi_\beta(0)=0$, that is, the
neuron is initialised at its resting potential. The parameters
$C_\mathrm{S}$ and $C_\mathrm{D}$ denote respectively the total
membrane capacitances of the soma and dendrite, and
$\mathcal{I}_\mathrm{S}(t)$ is the current supplied to the soma.

In the special case in which point currents $\mathcal{I}_1(t), \cdots,\mathcal{I}_n(t)$ act at distances $x_1,\cdots x_n$ from
the soma of the uniform cylinder, the corresponding coefficient
functions $\phi_0$ and $\phi_\beta$ satisfy
$$\label{ec2} \begin{array}{rcl} \ds\frac{d\phi_0}{dt} & = &\ds -\frac{e^{t/\tau}} {C_\mathrm{D}+C_\mathrm{S}}\,\Big[\, \mathcal{I}_\mathrm{S}(t)+\sum_{k=1}^n\;\mathcal{I}_k(t)\,\Big]\,,\\[10pt] \ds\frac{d\phi_\beta}{dt} & = & \ds-\frac{2e^{(1+\beta^2/L^2)t/\tau}} {C_\mathrm{D}+C_\mathrm{S}\cos^2\beta}\,\Big[\, \sum_{k=1}^n \;\mathcal{I}_k(t)\cos\beta\big(1-x_k/l\big) +\cos\beta\,\mathcal{I}_\mathrm{S}(t)\,\Big]\,. \end{array}$$